Friction Induces Anisotropic Propulsion in Sliding Magnetic Microtriangles

In viscous fluids, motile microentities such as bacteria or artificial swimmers often display different transport modes than macroscopic ones. A current challenge in the field aims at using friction asymmetry to steer the motion of microscopic particles. Here we show that lithographically shaped magnetic microtriangles undergo a series of complex transport modes when driven by a precessing magnetic field, including a surfing-like drift close to the bottom plane. In this regime, we exploit the triangle asymmetric shape to obtain a transversal drift which is later used to transport the microtriangle in any direction along the plane. We explain this friction-induced anisotropic sliding with a minimal numerical model capable to reproduce the experimental results. Due to the flexibility offered by soft-lithographic sculpturing, our method to guide anisotropic-shaped magnetic microcomposites can be potentially extended to many other field responsive structures operating in fluid media.

PDMS microwells were filled with the magnetic colloidal dispersion in ETPTA by sliding a 20muL droplet of it over the PDMS surface by tilting the mold to 45 • . After filling, the dispersion was reticulated in the wells overnight under a 254 nm hand-held UV lamp (NU4 KL, Benda Laborgeraete).
Experimental setup. The dynamics of the propelling particles are observed using an upright light microscope (Eclipse Ni, Nikon) equipped with a Charge-Coupled Device Camera (Scout scA640-74f, Basler) and different oil immersion objective (100× and 60×), depending on the degree of magnification required. We mount on the microscope stage a set of custom made magnetic coils arranged to apply time dependent magnetic fields. The coils that generate rotating field are driven by a power amplifier (IMG STA-800, stage line) which is controlled via a wave generator (Aim-TTi TGA1244). We obtain a rotating magnetic field in a given plane by passing through two perpendicular coils two sinusoidal current with 90 • phase shift. Further, a static field is obtained by using a DC power supplier (TTi El 302).

Section S2: Numerical simulation
We provide here a detailed account of the terms of Eq. ?? which determine the dynamics of the microtriangle. F g i = −mgẑ is the gravitational force. F LJ i (z i ) accounts for the steric interaction of a bead located a distance z i above the bounding planar solid wall. The steric potential is the Weeks-Chandler-Andersen (WCA) potential, which consists in the repulsive part of the Lennard-Jones potential. To calculate F m i we assume that the magnetic torque is applied to the centre of mass of the triangle at position r CM , As the torque does not induce any net force on the triangle, we add the extra constraint F m i = 0, which allows to rewrite Eq 1 as We also impose that the torque does not produce any local tensions along the sides of the triangle.
At each step, we solve numerically the system of equations by performing a lower-upper (LU) decomposition combined with a backward and a forward substitution algorithm. 2 Additionally, we ensure the separation between particles remains fixed by means of the MILC SHAKE algorithm.
in terms of the total force each bead is subject to, The first term in the right side of Eq. 3 corresponds to the self interaction contribution of the particle with its own image on the stationary bounding wall. The tensor ∆ν captures this interaction ∆ν = ν(1 −ẑẑ) + 2νẑẑ, a z i , a is the hydrodynamic radius of the bead, and z i its z coordinate distance to the stationary bounding wall. 4 The second term provides the cross hydrodynamic interactions between different beads: G(r i , r j ) ≡ G ij which takes into account the hydrodynamic flux contribution between a bead j, its image, and bead i. Due to this velocity field, particles experience an hydrodynamic drag that can be calculated as , so that the final expression for the force is Here we have considered an asymmetric frictionγ, which takes into account the difference in friction of beads when they move parallel to the plane of the triangle, or perpendicular to it. The termsnn andpp are tensors that determine the hydrodynamic friction normal (n) or perpendicular (p) to the triangle plane. The scalars γ ⊥ 0 , γ ∥ 0 denote the bead friction perpendicular and parallel to the triangle plane. This difference in friction accounts for the planar geometry of the triangle.
The tensor pp defines the plane of the triangle,pp = 1 −nn. In the limit γ ⊥ 0 =γ ∥ 0 , Eq. 4 reduces to the scalar form of the friction tensor for spherical beads. We consider the far field hydrodynamic coupling between beads and consider the Blake-Green expression for G ij , ? which takes into account the hydrodynamic interaction between beads in the presence of a stationary plane at z = 0.
Using the characteristic length of the triangle, r c , and the characteristic relaxation time τ = γ 0 ∥ r 2 c /|m||B|, one can express Eq. ?? in dimensionless form as where t a = m γ is the inertial time,r i = r i /r c , andF m i ≡ ∥F m i ||r|/|m||B|, and the factor a/r c compares the thickness to the size of the triangle.
The forceF g i = −ξê z , with ξ ≡ r c mg/|m||B| accounts for the relative magnitude of the gravitational field compared with the applied magnetic torque. A large torque compared to the We integrate Eq. 5 using an implicit, two step Velocity-Verlet algorithm in matrix notation to deal with the coupling introduced by the tensorial friction.
Section S3: Supporting video files